12.3 Diffusion Monte Carlo 389and that after the last branching step their actual number isM, then we adjustETas
ET=E 0 +αln(
M ̃
M
)
(12.55)
whereE 0 is close to the ground state energy (our ‘best estimate’), andαis some
small parameter.
In an algorithmic form, the resulting procedure can be presented as follows:
Put the walkers at random positions in configurational space;
REPEAT
FOR all walkers DO
Shift walker from its positionRto a new positionR′
according to the Gaussian transition probability(12.24);
Evaluateq=exp{−
τ[V(R′)−
τET]};
Eliminate the walker or create new ones atR′,
depending ons=q+r, whereris random,
uniform between 0 and 1;
END FOR;
UpdateET;
UNTIL finished.
The major difference with the variational Monte Carlo method described in the
previous section is that the present method does not rely on a trial function and
therefore the results have no systematic error due to the trial function being (in
general) not exact. There is, however, an error due to the fact that we have split the
time-evolution operator into two parts, one depending on the kinetic energy and
the other on the potential, by neglecting CBH commutators. By reducing
τwe
can make this error arbitrarily small, but the convergence speed will be reduced
accordingly. In Section 12.3.3, we shall describe a Metropolis algorithm to correct
for the discretisation error.
The population itself should represent the ground state wave function. For a one-
dimensional problem (or a radially symmetric three-dimensional problem) this can
be checked by constructing a histogram in which we record the frequencies with
which the various positions are occupied. Below we shall give some results of DMC
simulations for the harmonic oscillator and the helium atom.
The DMC procedure outlined here might fail in some cases. The distribution
of walkers can only represent a density which is positive everywhere. Therefore,
it can sample the ground wave function only if the latter is everywhere positive.
Fortunately, the ground state of a boson system is indeed everywhere positive.
However, for fermions this is no longer the case. Moreover, the Green’s function is
no longer positive in that case and it is not clear how to perform the diffusion, as